Converting recurring decimals to fractions

Converting recurring decimals into fractions is more difficult than converting terminating decimals, but there’s an ordered process that makes more sense with an example.

Examples

Example: Convert 0.\overline6 to a fraction

• let x = 0.\overline6 (the recurring decimal we want to convert)
• multiply both sides by 10 (since there’s 1 digit in the repeating part, so we use 10^1):
  • 10x = 6.\overline6
• subtract the original equation from this new equation:
  • 10x - x = 6.\overline6 - 0.\overline6
  • 9x = 6
• solve for x:
  • x = \frac{6}{9}
  • x = \frac{2}{3} (simplified)
• Answer: 0.\overline6 = \frac{2}{3}

Example: Convert 0.\overline{27} to a fraction

• let x = 0.\overline{27}
• multiply both sides by 100 (since there are 2 digits in the repeating part, so we use 10^2):
  • 100x = 27.\overline{27}
• subtract the original equation from this new equation:
  • 100x - x = 27.\overline{27} - 0.\overline{27}
  • 99x = 27
• solve for x:
  • x = \frac{27}{99}
  • x = \frac{3}{11} (simplified)
• Answer: 0.\overline{27} = \frac{3}{11}

Example: Convert 0.1\overline{3} to a fraction

• let x = 0.1\overline{3}
• multiply both sides by 10 (to move past the non-repeating part):
  • 10x = 1.\overline{3}
• multiply both sides by 10 again (to move past the repeating part):
  • 100x = 13.\overline{3}
• subtract the first new equation from the second new equation:
  • 100x - 10x = 13.\overline{3} - 1.\overline{3}
  • 90x = 12
• solve for x:
  • x = \frac{12}{90}
  • x = \frac{2}{15} (simplified)
• Answer: 0.1\overline{3} = \frac{2}{15}

Example: Convert 2.4\overline{56} to a fraction

• let x = 2.4\overline{56}
• multiply both sides by 10 (to move past the non-repeating part):
  • 10x = 24.\overline{56}
• multiply both sides by 100 (to move past the repeating part):
  • 1000x = 2456.\overline{56}
• subtract the first new equation from the second new equation:
  • 1000x - 10x = 2456.\overline{56} - 24.\overline{56}
  • 990x = 2432
• solve for x:
  • x = \frac{2432}{990}
  • x = \frac{1216}{495} (simplified)
• Answer: 2.4\overline{56} = \frac{1216}{495}

Example: Convert 7.1\overline{857} to a fraction

• let x = 7.1\overline{857}
• multiply both sides by 10 (to move past the non-repeating part):
  • 10x = 71.\overline{857}
• multiply both sides by 1000 (to move past the repeating part):
  • 10000x = 71857.\overline{857}
• subtract the first new equation from the second new equation:
  • 10000x - 10x = 71857.\overline{857} - 71.\overline{857}
  • 9990x = 71786
• solve for x:
  • x = \frac{71786}{9990}
  • x = \frac{35893}{4995} (simplified)
• Answer: 7.1\overline{857} = \frac{35893}{4995}